The continuous spontaneous localization (CSL) model is a spontaneous collapse model in quantum mechanics, proposed in 1989 by Philip Pearle. [1] and finalized in 1990 Gian Carlo Ghirardi, Philip Pearle and Alberto Rimini. [2]
The most widely studied among the dynamical reduction (also known as collapse) models is the CSL model. [1] [2] [3] Building on the Ghirardi-Rimini-Weber model, [4] the CSL model describes the collapse of the wave function as occurring continuously in time, in contrast to the Ghirardi-Rimini-Weber model.
Some of the key features of the model are: [3]
The CSL dynamical equation for the wave function is stochastic and non-linear:
Here is the Hamiltonian describing the quantum mechanical dynamics, is a reference mass taken equal to that of a nucleon, , and the noise field has zero average and correlation equal to
where denotes the stochastic average over the noise. Finally, we write
where is the mass density operator, which reads
where and are, respectively, the second quantized creation and annihilation operators of a particle of type with spin at the point of mass . The use of these operators satisfies the conservation of the symmetry properties of identical particles. Moreover, the mass proportionality implements automatically the amplification mechanism. The choice of the form of ensures the collapse in the position basis.
The action of the CSL model is quantified by the values of the two phenomenological parameters and . Originally, the Ghirardi-Rimini-Weber model [4] proposed s at m, while later Adler considered larger values: [5] s for m, and s for m. Eventually, these values have to be bounded by experiments.
From the dynamics of the wave function one can obtain the corresponding master equation for the statistical operator :
Once the master equation is represented in the position basis, it becomes clear that its direct action is to diagonalize the density matrix in position. For a single point-like particle of mass , it reads
where the off-diagonal terms, which have , decay exponentially. Conversely, the diagonal terms, characterized by , are preserved. For a composite system, the single-particle collapse rate should be replaced with that of the composite system
where is the Fourier transform of the mass density of the system.
Contrary to most other proposed solutions of the measurement problem, collapse models are experimentally testable. Experiments testing the CSL model can be divided in two classes: interferometric and non-interferometric experiments, which respectively probe direct and indirect effects of the collapse mechanism.
Interferometric experiments can detect the direct action of the collapse, which is to localize the wavefunction in space. They include all experiments where a superposition is generated and, after some time, its interference pattern is probed. The action of CSL is a reduction of the interference contrast, which is quantified by the reduction of the off-diagonal terms of the statistical operator [6]
where denotes the statistical operator described by quantum mechanics, and we define
Experiments testing such a reduction of the interference contrast are carried out with cold-atoms, [7] molecules [6] [8] [9] [10] and entangled diamonds. [11] [12]
Similarly, one can also quantify the minimum collapse strength to solve the measurement problem at the macroscopic level. Specifically, an estimate [6] can be obtained by requiring that a superposition of a single-layered graphene disk of radius m collapses in less than s.
Non-interferometric experiments consist in CSL tests, which are not based on the preparation of a superposition. They exploit an indirect effect of the collapse, which consists in a Brownian motion induced by the interaction with the collapse noise. The effect of this noise amounts to an effective stochastic force acting on the system, and several experiments can be designed to quantify such a force. They include: [13]
where is the vacuum dielectric constant and is the light speed. This prediction of CSL can be tested [18] [19] [20] [21] by analyzing the X-ray emission spectrum from a bulk Germanium test mass.
The CSL model consistently describes the collapse mechanism as a dynamical process. It has, however, two weak points.
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